Connecting the Concepts (8.1-8.4) Rational Expressions The operations sign should be noted when combining rational expressions since the process of adding and subtracting rational expressions is significantly different from multiplying and dividing them. Need Common Denominator? Operation Multiplication No Division No Do’s 2 2 Multiply: m m 6 2m 5m 3 2m 4 m3 m 2 m 6 2 m 2 5m 3 2m 4 m3 m 2 m 3 2m 3 m 1 2 m 2 m 3 2m 3 m 1 Procedure -Factor all numerators and denominators and simplify, if possible. -Write the factored form of the product. -Multiply the first rational expression with reciprocal of the second rational expression. (going back to multiplication) -Factor all numerators and denominators and then simplify, if possible. -Write the factored form of the product Tips & Cautions -Do not carry out the multiplication at the beginning. -Do not multiply the remaining factors after simplifying but write the product in factored form. Leave the final expression in factored form. -Begin by rewriting as a multiplication before factoring, simplifying. Don’ts m 2 m 6 2m 2 5m 3 2m 4 m3 2m 4 5m 3 3m 2 2m 3 5m 2 3m 12m 2 30m 18 2m 2 6m 4m 12 2m 4 7m 3 4m 2 27 m 18 2m 2 2m 12 2 It is not necessary to carry out these multiplications. 1 Need Common Denominator? Operation Addition Yes Subtraction Yes Procedure -If necessary, write equivalent rational expressions with least common denominator. -Add numerators and keep common denominators. -Simplify, if possible. -If necessary, write equivalent rational expressions with least common denominator. -Subtract the numerators and keep common denominators. -Simplify, if possible. Do’s x x 5x 6 2 x 3x 5 x2 2x 3 x 3 2 2 2 Factoring the denominators to find the LCD. LCD is x -Do not simplify after writing with the LCD. Instead, simplify, if possible, after subtracting the numerators. - Do not divide out terms. -Use parentheses around the numerator being subtracted. Don’ts x 3x 2 x 2 x 3 x 1 x 2 x 1 x 3 x 2 x 2 x 3 x 1 x 1 x 2 x 3 x -Do not simplify after writing with the LCD. Instead, simplify, if possible, after adding the numerators. -Do not divide out terms. 2 2 x Tips & Cautions x 2 x 3 x 1 . Writing equivallent expression with the LCD by myltiplying a form of 1. 2x 6 x x 3 x 3 x 1 5 x 3 x 5 x 3 x 1 x 2 x 3 x 1 x 2 x 3 x 1 x 2 x 2x 6 x 2 x 3 x 1 x 2 Subtracting numerators. Don't forget the parenthesis! x 2x 6 x 2 x 3 x 1 x 2 x6 x 2 x 3 x 1 x 2 x 3 x 2 x 3 x 1 x 3 x 3 x 1 Combining like terms in the numerator. Simplifying by factoring and removing a factor equal to 1. 4 z 9 3z 8 4z 3z 4 z 9 3z 8 4 z 3z 4 z 9 3z 8 z z 1 z 4 z 9 3z 8 4z 3z 4 z 9 3z 8 4z 3z 9 8 9 8 1 2 Math 31 (Activity # ____) Your Name: ___________________ Team Member #1__________________ Team Member #2.______________ Team Member #3__________________ Directions: Work collaboratively as a team to complete this activity. Look at the errors of each of the following two problems present on page 2. Work as a team to find the right way to approach the problem and present the solution step-by-step in an organized fashion. 4 z 9 3z 8 x 2 3x 5 2) 1) 2 4z 3z x 2x 3 x 3 Now try the following problems: 3) Multiply: m 2 m 6 2m 2 5m 3 2m 4 m3 4) Subtract: x x2 5x 6 2 x 2 3x 2 3 Mixed Review: Perform the indicated operation and, if possible, simplify. 8 5 2 3 9t 6t 6) 8 5 2 3 9t 6t 7) a3 a3 15a 3a 2 8) a3 a3 15a 3a 2 9) 3 2 x4 4 x 10) x 2 16 x2 x 2 x x 2 5x 4 11) 3u 2 3 4u 4 4 3 12) 3u 2 3 4u 4 4 3 5) Answers: 1) 4) 7) 10) x 3 11 x 2 x 15 x 3x 1x 3 x 3 x 3x 1 a 5 xx 4 x 1x 1 2) 5) 8) 11) 5 12 z 3) 16 15t 3 18t a 5a 3 15 a 2 9u 1 16 6) 9) 12) 2m 3m 1 2 20 27 t 5 5 x4 9u 2 16u 25 12 4